code: def left_integral3D(facets, index, expr, vertices, hp_param, degree):
      """Computes the left integral of Eq 10 in Chin et al.
  
      Explanation
      ===========
  
      For the 3D case, this is the sum of the integral values over constituting
      line segments of the face (which is accessed by facets[index]) multiplied
      by the distance between the first point of facet and that line segment.
  
      Parameters
      ==========
  
      facets :
          List of faces of the 3-Polytope.
      index :
          Index of face over which integral is to be calculated.
      expr :
          Input polynomial.
      vertices :
          List of vertices that constitute the 3-Polytope.
      hp_param :
          The hyperplane parameters of the face.
      degree :
          Degree of the ``expr``.
  
      Examples
      ========
  
      >>> from sympy.integrals.intpoly import left_integral3D
      >>> cube = [[(0, 0, 0), (0, 0, 5), (0, 5, 0), (0, 5, 5), (5, 0, 0),\
                   (5, 0, 5), (5, 5, 0), (5, 5, 5)],\
                   [2, 6, 7, 3], [3, 7, 5, 1], [7, 6, 4, 5], [1, 5, 4, 0],\
                   [3, 1, 0, 2], [0, 4, 6, 2]]
      >>> facets = cube[1:]
      >>> vertices = cube[0]
      >>> left_integral3D(facets, 3, 1, vertices, ([0, -1, 0], -5), 0)
      -50
      """
      value = S.Zero
      facet = facets[index]
      x0 = vertices[facet[0]]
      facet_len = len(facet)
      for i, fac in enumerate(facet):
          side = (vertices[fac], vertices[facet[(i + 1) % facet_len]])
          value += distance_to_side(
              x0, side, hp_param[0]
          ) * lineseg_integrate(facet, i, side, expr, degree)
      return value
  
imports: from functools import cmp_to_key
  from sympy.abc import x, y, z
  from sympy.core import S, diff, Expr, Symbol
  from sympy.core.sympify import _sympify
  from sympy.geometry import Segment2D, Polygon, Point, Point2D
  from sympy.polys.polytools import LC, gcd_list, degree_list, Poly
  from sympy.simplify.simplify import nsimplify
label_desc: Use image node_id 12 for calling a global function with example usage: left_integral3D(facets, index, expr, vertices, hp_param, degree) and returns: value